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{{main|Fractional quantum mechanics}}
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{{Quantum mechanics|cTopic=Advanced topics}}


The '''fractional Schrödinger equation''' is a fundamental equation of [[fractional quantum mechanics]]. It was discovered by [[Nick Laskin]] (1999) as a result of extending the [[Feynman path integral]], from the Brownian-like to Lévy-like quantum mechanical paths. The term ''fractional Schrödinger equation'' was coined by Nick Laskin.<ref>N. Laskin, (2000), [http://dx.doi.org/10.1016/S0375-9601(00)00201-2  Fractional Quantum Mechanics and Lévy Path Integrals. ''Physics Letters'' 268A, 298-304].</ref>
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==Fundamentals==
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The fractional Schrödinger equation in the form originally obtained by [[Nick Laskin]] is:<ref>N. Laskin, (2002), [http://pre.aps.org/abstract/PRE/v66/i5/e056108 Fractional Schrödinger equation, ''Physical Review'' E66, 056108 7 pages]. '' (also available online: http://arxiv.org/abs/quant-ph/0206098)</ref>
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{{Equation box 1
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|indent =:
|title=
|equation = <math>i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_\alpha (-\hbar
^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)+V(\mathbf{r},t)\psi (\mathbf{r},t)</math>
|cellpadding= 5
|border
|border colour = #0073CF
|background colour=#F5FFFA}}


*'''r''' is the 3-dimensional [[position vector]],
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*''ħ'' is the reduced [[Planck constant]],  
*''ψ''('''r''', ''t'') is the [[wavefunction]], which is the quantum mechanical probability amplitude for the particle to have a given position '''r''' at any given time ''t'',  
*''V''('''r''', ''t'') is a [[potential energy]],  
*Δ = &part;<sup>2</sup>/&part;'''r'''<sup>2</sup> is the [[Laplace operator]].


Further,
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*''D<sub>α</sub>'' is a scale constant with [[dimensional analysis|physical dimension]] [D<sub>α</sub>] = [energy]<sup>1 − ''α''</sup>·[length]<sup>''α''</sup>[time]<sup>−''α''</sup>, at ''α''&nbsp;=&nbsp;2, ''D''<sub>2</sub> =1/2''m'', where ''m'' is a particle mass,
*the operator (−''ħ''<sup>2</sup>Δ)<sup>''α''/2</sup> is the 3-dimensional fractional quantum Riesz derivative defined by (see, Ref.[2]);
 
::<math>
(-\hbar ^2\Delta )^{\alpha /2}\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar
)^3}\int d^3pe^{i\frac{\mathbf{pr}}\hbar }|\mathbf{p}|^\alpha \varphi (
\mathbf{p},t),
</math>
 
Here, the wave functions in the [[position and momentum space]]s; <math>\psi(\mathbf{r},t)</math> and <math> \varphi (\mathbf{p},t)</math> are related each other by the 3-dimensional [[Fourier transform]]s:
 
:<math>
\psi (\mathbf{r},t)=\frac 1{(2\pi \hbar )^3}\int d^3pe^{i \mathbf{p}\cdot\mathbf{r}/\hbar}\varphi (\mathbf{p},t),\qquad \varphi (\mathbf{p},t)=\int d^3re^{-i
\mathbf{p}\cdot\mathbf{r}/\hbar }\psi (\mathbf{r},t).
</math>
 
The index ''α'' in the fractional Schrödinger equation is the Lévy index, 1&nbsp;<&nbsp;''α''&nbsp;≤&nbsp;2. Thus, the fractional Schrödinger equation includes a space [[derivative]] of fractional order ''α'' instead of the second order (''α''&nbsp;=&nbsp;2) space derivative in the standard [[Schrödinger equation]]. Thus, the fractional Schrödinger equation is a [[fractional differential equation]] in accordance with modern terminology.<ref>S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional
Integrals and Derivatives, Theory and Applications ~Gordon
and Breach, Amsterdam, 1993</ref> This is the main point of the term ''fractional Schrödinger equation'' or a more general term [[fractional quantum mechanics]].<ref>N. Laskin, (2000), [http://pre.aps.org/abstract/PRE/v62/i3/p3135_1 Fractional Quantum Mechanics, ''Physical Review'' E62, 3135-3145]. '' (also available online: http://arxiv.org/abs/0811.1769)</ref> At ''α''&nbsp;=&nbsp;2 fractional Schrödinger equation becomes the well-known [[Schrödinger equation]].
 
The fractional Schrödinger equation has the following [[operator (physics)|operator]] form
 
{{Equation box 1
|indent =:
|title=
|equation = <math>
i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=\widehat{H}_\alpha
\psi (\mathbf{r},t)</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #ECFCF4}}
 
where the fractional Hamilton operator <math>\widehat{H}_\alpha </math> is given by
 
: <math>
\widehat{H}_\alpha =D_\alpha (-\hbar ^2\Delta )^{\alpha /2}+V(\mathbf{r},t).
</math>
 
The [[Hamilton operator]], <math>\widehat{H}_\alpha </math> corresponds to the [[classical mechanics]] [[Hamiltonian function]]
 
: <math>
H_\alpha (\mathbf{p},\mathbf{r})=D_\alpha |\mathbf{p}|^\alpha +V(\mathbf{r},t),
</math>
 
where '''p''' and '''r''' are the momentum and the position vectors respectively.
 
===Time-independent fractional Schrödinger equation===
 
The special case when the Hamiltonian <math>H_\alpha </math> is independent of time
: <math>
H_\alpha =D_\alpha (-\hbar ^2\Delta )^{\alpha /2}+V(\mathbf{r}),
</math>
is of great importance for physical applications.
It is easy to see that in this case there exist the special solution of the fractional Schrödinger equation
: <math>
\psi (\mathbf{r},t)=e^{-(i/\hbar )Et}\phi (\mathbf{r}),
</math>
where <math>\phi (\mathbf{r})</math> satisfies
:<math>
H_\alpha \phi (\mathbf{r}) = E\phi (\mathbf{r}), 
</math>
 
or
 
:<math>
D_\alpha (-\hbar ^2\Delta )^{\alpha /2}\phi (\mathbf{r})+V(\mathbf{r})\phi (
\mathbf{r})=E\phi (\mathbf{r}).
</math>
 
This is the '''time-independent fractional Schrödinger equation'''.
 
Thus, we see that the [[wave function]] <math>\psi (\mathbf{r},t)</math> oscillates with a definite frequency. In [[classical physics]] the frequency corresponds to the energy. Therefore, the quantum mechanical state has a definite energy ''E''.
The probability to find a particle at <math>\mathbf{r}</math> is the absolute square of the wave function <math>| \psi (\mathbf{r},t) |^2 .</math>
Because of time-independent fractional Schrödinger equation this is equal to <math>| \phi (\mathbf{r})|^2 </math>  and does not depend upon the time.
That is, the probability of finding the particle at <math>\mathbf{r}</math> is independent of the time. One can say that the system is in a stationary
state. In other words, there is no variation in the probabilities as a function of time.
 
===Probability current density===
 
The [[continuity equation#Quantum mechanics|continuity equation]] for probability current and density follows from the fractional Schrödinger equation:
 
:<math>
\frac{\partial \rho (\mathbf{r},t)}{\partial t}+\nabla \cdot \mathbf{j}(
\mathbf{r},t)=0, 
</math>
 
where <math>\rho (\mathbf{r},t)=\psi ^{\ast }(\mathbf{r},t)\psi (\mathbf{r},t)</math>
is the quantum mechanical probability density and the vector <math>\mathbf{j}(\mathbf{r},t)</math>
can be called by the fractional probability current density vector
 
:<math>
\mathbf{j}(\mathbf{r},t)=\frac{D_\alpha \hbar }i\left( \psi ^{*}(\mathbf{r}
,t)(-\hbar ^2\Delta )^{\alpha /2-1}\mathbf{\nabla }\psi (\mathbf{r},t)-\psi (
\mathbf{r},t)(-\hbar ^2\Delta )^{\alpha /2-1}\mathbf{\nabla }\psi ^{*}(
\mathbf{r},t)\right) , 
</math>
 
where we use the notation (see also [[matrix calculus#Scope|matrix calculus]]): <math>
\mathbf{\nabla =\partial /\partial r}
</math>.
 
Introducing the [[momentum operator]] <math>\widehat{\mathbf{p}}=\frac{\hbar }{i}
\frac{\partial }{\partial \mathbf{r}}</math> we can write the vector <math>\mathbf{j}</math>
in the form (see, Ref.[2])
 
:<math>
\mathbf{j=}D_{\alpha }\left( \psi (\widehat{\mathbf{p}}^{2})^{\alpha /2-1}
\widehat{\mathbf{p}}\psi ^{\ast }+\psi ^{\ast }(\widehat{\mathbf{p}}^{\ast
2})^{\alpha /2-1}\widehat{\mathbf{p}}^{\ast }\psi \right). 
</math>
 
This is fractional generalization of the well-known equation for probability current density
vector of standard quantum mechanics (see, Ref.[7]).
 
====Velocity operator====
 
The quantum mechanical velocity operator <math>\widehat{\mathbf{v}}</math> is defined as follows:
 
:<math>
\widehat{\mathbf{v}}=\frac{i}{\hbar }(H_{\alpha }\widehat{\mathbf{r}}\mathbf{
-}\widehat{\mathbf{r}}H_{\alpha }),
</math>
 
Straightforward calculation results in (see, Ref.[2])
:<math>
\widehat{\mathbf{v}}=\alpha D_{\alpha }|\widehat{\mathbf{p}}^{2}|^{\alpha
/2-1}\widehat{\mathbf{p}}\,.
</math>
 
Hence,
 
:<math>
\mathbf{j=}\frac{1}{\alpha }\left( \psi \widehat{\mathbf{v}}\psi ^{\ast
}+\psi ^{\ast }\widehat{\mathbf{v}}\psi \right) ,\qquad 1<\alpha \leq 2.
</math>
 
To get the [[probability current]] density equal to 1 (the current when one
particle passes through unit area per unit time) the wave function of a free
particle has to be normalized as
 
:<math>
\psi (\mathbf{r},t)=\sqrt{\frac{\alpha }{2\mathrm{v}}}\exp \left[\frac{i}{\hbar }(
\mathbf{p}\cdot\mathbf{r}-Et)\right],\qquad E=D_{\alpha }|\mathbf{p}|^{\alpha
},\qquad 1<\alpha \leq 2, 
</math>
 
where <math>\mathrm{v}</math>  is the particle [[velocity]], <math>\mathrm{v}=\alpha D_{\alpha
}p^{\alpha -1}</math>.
 
Then we have
 
:<math>
\mathbf{j=}\frac{\mathbf{v}}{\mathrm{v}},\qquad \mathbf{v}=\alpha D_{\alpha
}|\mathbf{p}^{2}|^{\frac{\alpha }{2}-1}\mathbf{p,} 
</math>
 
that is, the vector <math>\mathbf{j}</math> is indeed the [[unit vector]].
 
== Physical applications ==
 
===Fractional Bohr atom===
 
{{main|Bohr atom}}
 
When <math>V(\mathbf{r})</math> is the potential energy of [[hydrogenlike atom]],
 
: <math>
V(\mathbf{r})=-\frac{Ze^{2}}{|\mathbf{r}|},
</math>
 
where ''e'' is the [[electron charge]] and ''Z'' is the [[atomic number]] of the hydrogenlike atom, (so ''Ze'' is the nuclear charge of the atom), we come to following fractional [[eigenvalue]] problem,
 
: <math>
D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}\phi (\mathbf{r})-\frac{Ze^{2}}{|
\mathbf{r|}}\phi (\mathbf{r})=E\phi (\mathbf{r}).
</math>
 
This eigenvalue problem has first been solved in.<ref>N. Laskin, (2000), [http://chaos.aip.org/resource/1/chaoeh/v10/i4/p780_s1?isAuthorized=no Fractals and quantum mechanics. Chaos 10, 780-790]</ref>
 
Using the first [[Niels Bohr]] postulate yields
: <math>
\alpha
D_{\alpha }\left( \frac{n\hbar }{a_{n}}\right) ^{\alpha }=\frac{Ze^{2}}{a_{n}
},
</math>
 
and it gives us the equation for the [[Bohr radius]] of the fractional hydrogenlike atom
: <math>
a_{n}=a_{0}n^{\alpha /(\alpha -1)}. 
</math>
 
Here ''a''<sub>0</sub> is the fractional Bohr radius (the radius of the lowest, ''n'' = 1, Bohr orbit) defined as,
 
: <math>
a_{0}=\left( \frac{\alpha D_{\alpha }\hbar ^{\alpha }}{Ze^{2}}\right) ^{1/(\alpha -1)}. 
</math>
 
The [[energy levels]] of the fractional hydrogenlike atom are given by
 
: <math>
E_{n}=(1-\alpha )E_{0}n^{-\alpha/(\alpha -1)},\qquad 1<\alpha \leq 2,
</math>
 
where ''E''<sub>0</sub> is the [[binding energy]] of the electron in the lowest Bohr orbit
that is, the energy required to put it in a state with ''E'' = 0 corresponding to ''n'' = ∞,
 
: <math>
E_{0}=\left( \frac{Ze^{2}}{\alpha D_{\alpha }^{1/\alpha }\hbar }\right) ^{\alpha/(\alpha -1)}. 
</math>
 
The energy (''α''  − 1)''E''<sub>0</sub> divided by ''ħc'', (''α''  − 1)''E''<sub>0</sub>/''ħc'',  can be considered as fractional generalization of the
[[Rydberg constant]] of standard [[quantum mechanics]]. For ''α''&nbsp;=&nbsp;2 and ''Z'' = 1  the formula
<math>(\alpha -1)E_{0}/\hbar c</math> is transformed into
 
: <math>\mathrm{Ry}=me^{4}/2\hbar ^{3}c</math>,
which is the well-known expression for the [[Rydberg formula]].
 
According to the second [[Niels Bohr]] postulate, the frequency of radiation <math>\omega</math> associated with the transition, say, for example from the orbit ''m'' to the orbit ''n'', is,
 
: <math>
\omega =\frac{(1-\alpha )E_{0}}{\hbar }\left[ \frac{1}{n^{\frac{\alpha }{
\alpha -1}}}-\frac{1}{m^{\frac{\alpha }{\alpha -1}}}\right] 
</math>.
 
The above equations are fractional
generalization of the Bohr model. In the special Gaussian case, when (''α''&nbsp;=&nbsp;2) those equations give us the well-known results of the [[Bohr model]].<ref>N. Bohr, (1913), Phil. Mag. 26, 1, 476, 857</ref>
 
===The infinite potential well===
 
A particle in a one-dimensional well moves in a potential field <math>V({x})</math>, which is zero for
<math>-a\leq x\leq a</math> and which is infinite elsewhere,
 
: <math>
V(x)=\infty ,\qquad x<-a\qquad \qquad (\mathrm{i})
</math>
 
: <math>
V(x)=0,\quad -a\leq x\leq a\quad \quad \quad \ (\mathrm{ii}) 
</math>
 
: <math>
V(x)=\infty ,\qquad \ x>a\qquad \qquad \ (\mathrm{iii})
</math>
 
It is evident ''a priori'' that the [[energy spectrum]] will be discrete. The solution of the fractional Schrödinger equation for the stationary state with well-defined energy ''E'' is described by a wave function <math>\psi (x)</math>, which can be written as
 
:<math>
\psi (x,t)=\left(-i\frac{Et}{\hbar }\right)\phi(x)
</math>,
 
where <math>\phi (x)</math>,  is now time independent.
In regions (i) and (iii),
the fractional Schrödinger
equation can be satisfied only if we take <math>\phi (x)=0</math>. In the middle region
(ii), the time-independent fractional Schrödinger equation is
 
: <math>
D_\alpha (\hbar \nabla )^\alpha \phi (x)=E\phi (x). 
</math>
 
This equation defines the [[wave functions]] and the energy spectrum within region (ii), while outside
of the region (ii), x<-a and x>a, the wave functions  are zero. The wave function <math>\phi (x)</math> has to be continuous everywhere, thus we impose the boundary conditions <math>\phi (-a)=\phi (a)=0</math> for the solutions of the ''time-independent fractional Schrödinger equation'' (see, Ref.[5]). Then the solution in region (ii) can be written as
 
:<math>
\phi ^{\mathrm{even}}(x)=A\cos kx,\qquad \text{or}\qquad \phi ^{\mathrm{odd
}}(x)=A\sin kx,
</math>
 
where ''k'' is given by
 
:<math>
k=\frac 1\hbar (\frac E{D_\alpha })^{1/\alpha },\qquad 1<\alpha \leq 2.
</math>
 
The even (under reflection <math>x\rightarrow -x</math>) solution <math>\phi ^{\mathrm{even}}(x)</math> satisfies the boundary conditions if
 
:<math>
k=(2m-1)\frac \pi {2a},\quad \quad m=1,2,3,...
</math>
 
The odd (under reflection <math>x\rightarrow -x</math>) solution <math>\phi ^{\mathrm{odd}}(x)</math> satisfies the boundary conditions if
 
:<math>
k=\frac{m\pi }a,\quad \quad m=1,2,3,... 
</math>
 
It is easy to check that the normalized solutions are
 
:<math>
\phi _m^{\mathrm{even}}(x)=\frac 1{\sqrt{a}}\cos \left[ (m-\frac 12)\frac{
\pi x}a\right] ,
</math>
 
and
 
:<math>
\phi _m^{\mathrm{odd}}(x)=\frac 1{\sqrt{a}}\sin \frac{m\pi x}a.
</math>
 
Solutions <math>\phi ^{\mathrm{even}}(x)</math> and <math>\phi ^{\mathrm{odd}}(x)</math> have the
property that
 
:<math>
\int\limits_{-a}^{a}dx\phi _{m}^{\mathrm{even}}(x)\phi _{n}^{\mathrm{even}
}(x)=\int\limits_{-a}^{a}dx\phi _{m}^{\mathrm{odd}}(x)\phi _{n}^{\mathrm{odd}
}(x)=\delta _{mn},
</math>
 
where <math>\delta _{mn}</math> is the [[Kronecker symbol]] and
 
:<math>
\int\limits_{-a}^{a}dx\phi _{m}^{\mathrm{even}}(x)\phi _{n}^{\mathrm{odd}
}(x)=0.
</math>
 
The eigenvalues of the particle in an infinite potential well are (see, Ref.[5])
 
: <math>
E_n=D_\alpha \left( \frac{\pi \hbar }a\right) ^\alpha n^\alpha ,\qquad
\qquad n=1,2,3....,\qquad 1<\alpha \leq 2. 
</math>
 
It is obvious that in the Gaussian case (''α''&nbsp;=&nbsp;2) above equations are
transformed into the standard quantum mechanical equations for a [[particle in a box]] (for example, see
Eq.(20.7) in <ref>L.D. Landau and E.M. Lifshitz, Quantum mechanics (Non-relativistic Theory), Vol.3, Third Edition, Course of Theoretical Physics, Butterworth-Heinemann, Oxford, 2003</ref>)
 
The state of the lowest energy, the [[ground state]], in the infinite potential
well is represented by the <math>\phi _n^{\mathrm{even}}(x)</math> at ''n''=1,
: <math>
\phi _{\mathrm{ground}}(x)\equiv \phi _1^{\mathrm{even}}(x)=\frac 1{\sqrt{a}
}\cos \left(\frac{\pi x}{2a}\right),
</math>
 
and its energy is
 
: <math>
E_{\mathrm{ground}}=D_{\alpha }\left( \frac{\pi \hbar }{2a}\right) ^{\alpha
}. 
</math>
 
===Fractional quantum oscillator===
 
''Fractional quantum oscillator'' introduced by [[Nick Laskin]] (see, Ref.[2]) is the fractional quantum  mechanical model with the  [[Hamiltonian operator]] <math>H_{\alpha ,\beta }</math> defined as
 
:<math>
H_{\alpha,\beta}=D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}+q^{2}|\mathbf{
r}|^{\beta },\quad 1<\alpha \leq 2,\quad 1<\beta \leq 2, 
</math>,
 
where ''q'' is interaction constant.
 
The fractional Schrödinger equation for the wave
function <math>\psi (\mathbf{r},t)</math> of the fractional quantum oscillator is,
 
:<math>
i\hbar \frac{\partial \psi (\mathbf{r},t)}{\partial t}=D_{\alpha }(-\hbar
^{2}\Delta )^{\alpha /2}\psi (\mathbf{r},t)+q^{2}|\mathbf{r}|^{\beta }\psi (
\mathbf{r},t) 
</math>
 
Aiming to search for solution in form
 
:<math>
\psi (\mathbf{r},t)=e^{-iEt/\hbar }\phi (\mathbf{r}),
</math>
 
we come to the time-independent fractional Schrödinger equation,
 
:<math>
D_{\alpha }(-\hbar ^{2}\Delta )^{\alpha /2}\phi (\mathbf{r},t)+q^{2}|\mathbf{
r}|^{\beta }\phi (\mathbf{r},t)=E\phi (\mathbf{r},t). 
</math>
 
The Hamiltonian <math>H_{\alpha,\beta}</math> is the fractional
generalization of the 3D [[quantum harmonic oscillator]] Hamiltonian of standard quantum
mechanics.
 
====Energy levels of the 1D fractional quantum oscillator in semiclassical approximation====
 
The [[energy levels]] of 1D fractional quantum oscillator with the [[Hamiltonian function]] <math>
H_{\alpha}=D_{\alpha }|p|^{\alpha }+q^{2}|x|^{\beta }</math> can be found in semiclassical approximation.
 
We set the total energy equal to ''E'', so that
 
:<math>
E=D_{\alpha }|p|^{\alpha }+q^{2}|x|^{\beta }, 
</math>
 
whence
 
:<math>
|p|=\left( \frac{1}{D_{\alpha }}(E-q^{2}|x|^{\beta })\right)
^{1/\alpha }
</math>.
 
At the turning points <math>p=0</math>. Hence, the classical motion is
possible in the range <math>|x|\leq (E/q^{2})^{1/\beta }</math>.
 
A routine use of the [[Bohr-Sommerfeld quantization]] rule yields
 
:<math>
2\pi \hbar (n+\frac{1}{2})=\oint
pdx=4\int\limits_{0}^{x_{m}}pdx=4\int\limits_{0}^{x_{m}}D_{\alpha
}^{-1/\alpha }(E-q^{2}|x|^{\beta })^{1/\alpha }dx, 
</math>
 
where the notation <math>\oint  </math> means the integral over one complete period of
the classical motion and <math>x_{m}=(E/q^{2})^{1/\beta }</math> is the turning point
of classical motion.
 
To evaluate the integral in the right hand we introduce a new variable <math>y=x(E/q^{2})^{-1/\beta }</math>. Then we have
 
:<math>
\int\limits_0^{x_m}D_\alpha ^{-1/\alpha }(E-q^2|x|^\beta )^{1/\alpha }dx=\frac 1{D_\alpha ^{1/\alpha }q^{2/\beta }}E^{\frac 1\alpha +\frac 1\beta
}\int\limits_0^1dy(1-y^\beta )^{1/\alpha }.
</math>
 
The integral over ''dy'' can be expressed in terms of the [[Beta-function]],
 
:<math>
\int\limits_{0}^{1}dy(1-y^{\beta })^{1/\alpha }=\frac{1}{\beta }
\int\limits_{0}^{1}dzz^{\frac{1}{\beta }-1}(1-z)^{\frac{1}{\alpha }}=\frac{1
}{\beta }\Beta \left(\frac{1}{\beta },\frac{1}{\alpha }+1\right). 
</math>
 
Therefore
 
:<math>
2\pi \hbar (n+\frac 12)=\frac 4{D_\alpha ^{1/\alpha }q^{2/\beta }}E^{\frac
1\alpha +\frac 1\beta }\frac 1\beta \Beta\left(\frac 1\beta ,\frac 1\alpha +1\right).
</math>
 
The above equation gives the energy levels of stationary states for
the 1D fractional quantum oscillator (see, Ref.[2]),
 
:<math>
E_{n}=\left( \frac{\pi \hbar \beta D_{\alpha }^{1/\alpha }q^{2/\beta }}{2\Beta(
\frac{1}{\beta },\frac{1}{\alpha }+1)}\right) ^{\frac{\alpha \beta }{\alpha
+\beta }}\left(n+\frac{1}{2}\right)^{\frac{\alpha \beta }{\alpha +\beta }}.
</math>
 
This equation is generalization of the well-known [[energy levels]] equation of the
standard [[quantum harmonic oscillator]] (see, Ref.[7]) and
is transformed into it at ''α''&nbsp;=&nbsp;2 and ''β''&nbsp;=&nbsp;2.
It follows from this equation that at <math>\frac{1}{\alpha }+\frac{1}{\beta }=1</math> the energy levels are equidistant. When <math>1<\alpha \leq 2 </math> and <math>1<\beta \leq 2</math> the equidistant energy levels can be for ''α''&nbsp;=&nbsp;2 and ''β''&nbsp;=&nbsp;2 only. It means that the only standard quantum
harmonic oscillator has an [[equidistant]] [[energy spectrum]].
 
== See also ==
*[[Schrödinger equation]]
*[[Path integral formulation]]
*[[Relation between Schrödinger's equation and the path integral formulation of quantum mechanics]]
*[[Fractional calculus]]
*[[Quantum harmonic oscillator]]
 
== References ==
 
<references/>
<!---NO DELETING--->
*{{cite book|title=Fractional Calculus, An Introduction for Physicists|author=Richard Herrmann|year=2011|publisher=World Scientific|chapter=9|isbn=981 4340 24 3|url=http://books.google.co.uk/books?id=mPXzp1f7ycMC&pg=PA97&dq=fractional+schrodinger+equation&hl=en&sa=X&ei=rtWMUYSTBcXb0QXKzIHgAw&redir_esc=y#v=onepage&q=fractional%20schrodinger%20equation&f=false}}
 
*{{cite book|title=Fractional Dynamics: Recent Advances|author=J. Klafter, S.C. Lim, R. Metzler|year=2012|publisher=World Scientific|page=426|isbn=981-434-059-6|url=http://books.google.co.uk/books?id=wwfqHij4V2MC&dq=fractional+schrodinger+equation&source=gbs_navlinks_s}}
 
*{{cite book|title=Progress in Statistical Mechanics Research|author=J.S. Moreno|year=2008|publisher=Nova Publishers|page=10|isbn=160-456-028-2|url=http://books.google.co.uk/books?id=pJcWp7RaHqsC&pg=PA10&dq=fractional+schrodinger+equation&hl=en&sa=X&ei=rtWMUYSTBcXb0QXKzIHgAw&redir_esc=y#v=onepage&q=fractional%20schrodinger%20equation&f=false}}
 
*{{cite book|title=Nonlinear partial differential equations for scientists and engineers|author=L. Debnath|year=2005|edition=2nd|publisher=Springer|chapter=|pages=126–127|isbn=0-817-643-230|url=http://books.google.co.uk/books?id=fqI4kHP-5iMC&pg=PA126&dq=fractional+schrodinger+equation&hl=en&sa=X&ei=rtWMUYSTBcXb0QXKzIHgAw&redir_esc=y#v=onepage&q=fractional%20schrodinger%20equation&f=false}}
 
*{{cite book|title=Linear partial differential equations for scientists and engineers|author=T. Myint U., L. Debnath|year=2007|edition=4th|publisher=Springer|chapter=|page=520|isbn=0-817-645-608|url=http://books.google.co.uk/books?id=Zbz5_UvERIIC&pg=PA520&dq=fractional+schrodinger+equation&hl=en&sa=X&ei=4NiMUdy_OK340gWHl4D4CA&redir_esc=y#v=onepage&q=fractional%20schrodinger%20equation&f=false}}
 
*{{cite book|title=Issues in Applied Mathematics|author=|year=2012|publisher=Scholarly Editions|page=|isbn=146-496-507-2|url=http://books.google.co.uk/books?id=pl80j96H_o4C&pg=PT560&dq=fractional+schrodinger+equation&hl=en&sa=X&ei=rtWMUYSTBcXb0QXKzIHgAw&redir_esc=y}}
 
*{{cite book|title=Issues in General Physics Research|author=|year=2012|publisher=Scholarly Editions|chapter=8|page=|isbn=146-496-328-2|url=http://books.google.co.uk/books?id=aOdLpmAHdLoC&pg=PA1126&dq=fractional+schrodinger+equation&hl=en&sa=X&ei=rtWMUYSTBcXb0QXKzIHgAw&redir_esc=y#v=onepage&q=fractional%20schrodinger%20equation&f=false}}
 
*{{cite book|title=Fractional dynamics|author=V.E. Tarasov|year=2010|volume=0|series=Nonlinear physical science|publisher=Springer|chapter=19|page=|isbn=3-642-140-033|url=http://books.google.co.uk/books?id=Mc193Swt_JQC&pg=PA465&dq=Fractional+quantum+mechanics+N+Laskin&hl=en&sa=X&ei=zt-MUcPyMYv20gXNu4HYBA&ved=0CDEQ6AEwAA#v=onepage&q=Fractional%20quantum%20mechanics%20N%20Laskin&f=false}}
 
*{{cite book|title=Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering |author=J. Sabatier, O.P.Agrawal, J.A.T.Machado|year=2007|volume=|series=|publisher=Springer|chapter=|page=|isbn=1-402-060-424|url=http://books.google.co.uk/books?id=8lBgSDVXULwC&pg=PA126&dq=Fractional+quantum+mechanics+N+Laskin&hl=en&sa=X&ei=zt-MUcPyMYv20gXNu4HYBA&ved=0CFgQ6AEwCA#v=onepage&q=Fractional%20quantum%20mechanics%20N%20Laskin&f=false}}
 
*{{cite book|title=Fractional Dynamics and Control|author=D. Baleanu, J.A.T. Machado, A.C.J. Luo|year=2012|volume=|series=|publisher=Springer|chapter=17|page=|isbn=1-461-404-576|url=http://books.google.co.uk/books?id=8tpLwiLONDkC&pg=PA210&dq=Fractional+quantum+mechanics+N+Laskin&hl=en&sa=X&ei=UeGMUd_JK66r0gXpp4GQCw&ved=0CEQQ6AEwBDgK#v=onepage&q=Fractional%20quantum%20mechanics%20N%20Laskin&f=false}}
 
== Further reading ==
* [http://dx.doi.org/10.1063/1.2235026 Xiaoyi Guo and Mingyu Xu, Some physical applications of fractional Schrödinger equation. J. Math. Phys. 47, 082104 (2006).]
* [http://jmp.aip.org/jmapaq/v48/i4/p043502_s1?isAuthorized=no  S. Wang, M. Xu, Generalized fractional Schrödinger equation with space-time fractional derivatives J. Math. Phys. 48 (2007) 043502 ]
* [http://dx.doi.org/10.1063/1.4705268 Bayın, Selçuk Ş., On the consistency of the solutions of the space fractional Schrödinger equation. Journal of Mathematical Physics, Volume 53, Issue 4, pp. 042105-042105-9 (2012)]
* [http://dx.doi.org/10.1063/1.2749172 Jianping Dong, , Mingyu Xu, Some solutions to the space fractional Schrödinger equation using momentum representation method , J. Math. Phys. 48, 072105 (2007).]
* [http://www.sciencedirect.com/science/article/pii/S0022247X08003338 Jianping Dong, , Mingyu Xu, Space–time fractional Schrödinger equation with time-independent potentials, Journal of Mathematical Analysis and Applications Volume 344, Issue 2, Pages 1005–1017 (2008).]
* [http://pre.aps.org/abstract/PRE/v80/i2/e022103 A. Iomin, Fractional-time quantum dynamics. Phys. Rev. E 80, (2009) 022103.]
* [http://chaos.aip.org/resource/1/chaoeh/v10/i4/p780_s1?isAuthorized=no N. Laskin, Fractals and quantum mechanics. Chaos 10(2000) 780-790]
* [http://jmp.aip.org/jmapaq/v45/i8/p3339_s1?isAuthorized=no M. Naber, Time fractional Schrodinger equation. J. Math. Phys. 45 (2004) 3339-3352.] [http://arxiv.org/abs/math-ph/0410028 arXiv:math-ph/0410028]
* [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVM-4RPM7RC-1&_user=10&_coverDate=04%2F21%2F2008&_rdoc=1&_fmt=high&_orig=search&_sort=d&_docanchor=&view=c&_searchStrId=1400457817&_rerunOrigin=google&_acct=C000050221&_version=1&_urlVersion=0&_userid=10&md5=fd7815c9c95482bc8873a50573068777 V.E. Tarasov, Fractional Heisenberg equation. Phys. Lett. A 372 (2008) 2984-2988.]
* [http://jmp.aip.org/jmapaq/v49/i10/p102112_s1?bypassSSO=1 V.E. Tarasov, Weyl quantization of fractional derivatives. J. Math. Phys. 49 (2008) 102112.]
* [http://dx.doi.org/10.1063/1.3525976 Edmundo Capelas de Oliveira, Felix Silva Costa, and Jayme Vaz, Jr., The fractional Schrödinger equation for delta potentials, J. Math. Phys. 51, 123517 (2010).]
* [http://iopscience.iop.org/1751-8121/44/18/185303 E Capelas de Oliveira and Jayme Vaz Jr, "Tunneling in Fractional Quantum Mechanics" Journal of Physics A Volume 44 (2011) 185303.]
 
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Latest revision as of 21:52, 2 October 2014

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